Problem 35 - Algebra for College Students

Question

For Problems \(31-44\), solve each equation for the indicated variable. $$ I=\frac{100 M}{C} \text { for } M $$

Step-by-Step Solution

Answer

\(M = \frac{C \cdot I}{100}\)

Step 1: Understand the Equation

The equation given is \(I = \frac{100M}{C}\). Our task is to solve this equation for the variable \(M\). This means we need to express \(M\) in terms of the other variables, \(I\) and \(C\).

Step 2: Isolate \(M\)

To isolate \(M\), we first need to get rid of the fraction. We can do this by multiplying both sides of the equation by \(C\): \(C \cdot I = 100M\).

Step 3: Solve for \(M\)

Now that we have \(C \cdot I = 100M\), the final step is to divide both sides by 100 to solve for \(M\): \(M = \frac{C \cdot I}{100}\).

Key Concepts

Solving equations

In algebra, solving equations involves finding the value of a variable that makes the equation true. An equation is a statement asserting the equality of two expressions, and our goal is to determine the unknown quantity. This process requires logical reasoning and the application of properties of equality. Consider the equation in the exercise, \( I = \frac{100M}{C} \), where we are asked to solve for \( M \). Solving means writing \( M \) as an explicit function of the other variables, \( I \) and \( C \).

• Identify what you are solving for — in this case, the variable \( M \).
• You may need to perform operations such as addition, subtraction, multiplication, division, or a combination to manipulate the equation into a solvable form.
• Preserve the balance of the equation by performing the same operation to both sides.

Breaking down equations systematically allows for the accurate determination of unknowns and the manipulation of mathematical relationships.

Isolating variables

Isolating a variable means rearranging an equation so that the variable you want to solve for stands alone on one side of the equation. This process is crucial in solving equations and involves reversing operations attached to the variable being isolated.To isolate \( M \) in the given equation \( I = \frac{100M}{C} \), we need to eliminate the components linked to \( M \):

• First, multiply both sides by \( C \) to clear the fraction: \( C \cdot I = 100M \). This operation "undoes" the division by \( C \), placing \( M \) in a linear term.
• Then, divide by 100, the coefficient of \( M \), resulting in \( M = \frac{C \cdot I}{100} \).

This systematic isolation of variables facilitates problem-solving by organizing the equation to visibly show what changes need to occur to solve for a given variable.

Fraction elimination

Fractions can complicate equations, making fraction elimination a key strategy in algebra. By removing fractions, we simplify equations, making them easier to handle and understand.In the equation \( I = \frac{100M}{C} \), the fraction \( \frac{100M}{C} \) is present. Here's how to eliminate it:

• Multiply every term by the denominator (\( C \)) to "undo" the division and obtain\( C \cdot I = 100M \).
• This step removes the fraction, converting the equation to a simpler form.

After elimination, proceed with standard operations like division or multiplication to isolate the desired variable. This technique is foundational in algebra, promoting a clear path toward solving equations.